$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of functions $\mathbb{R} \to \mathbb{C}$ which are continuous and of bounded variation.

Is $\BV(\mathbb{R})$ Arens regular?

Here's how far I've gotten until now:

• I have not seen the question mentioned in any of the literature so far. In particular, it doesn't seem to be in Palmer's book.

• Since $C^*$-algebras are Arens regular, it would be enough to realize $\BV(\mathbb{R})$ as a subalgebra of a $C^*$-algebra, or equivalently to represent it on a Hilbert space.

• Characterizing the dual space $\BV(\mathbb{R}^n)^*$ is an open problem, but I'm not sure if this includes the case $n = 1$. If so, I'm hoping that my question might still have a known answer that doesn't depend on a characterization of the dual first.

$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of functions $\mathbb{R} \to \mathbb{C}$ which are continuous and of bounded variation.

Is $\BV(\mathbb{R})$ Arens regular?

Here's how far I've gotten until now:

• I have not seen the question mentioned in any of the literature so far. In particular, it doesn't seem to be in Palmer's book.

• Since $C^*$-algebras are Arens regular, it would be enough to realize $\BV(\mathbb{R})$ as a closed subalgebra of a $C^*$-algebra, or equivalently to represent it on a Hilbert space.

• Characterizing the dual space $\BV(\mathbb{R}^n)^*$ is an open problem, but I'm not sure if this includes the case $n = 1$. If so, I'm hoping that my question might still have a known answer that doesn't depend on a characterization of the dual first.

A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $BV(\mathbb{R})$ denote the Banach algebra of functions $\mathbb{R} \to \mathbb{C}$ which are continuous and of bounded variation.

Is $BV(\mathbb{R})$ Arens regular?

Here's how far I've gotten until now:

• I have not seen the question mentioned in any of the literature so far. In particular, it doesn't seem to be in Palmer's book.

• Since $C^*$-algebras are Arens regular, it would be enough to realize $BV(\mathbb{R})$ as a subalgebra of a $C^*$-algebra, or equivalently to represent it on a Hilbert space.

• Characterizing the dual space $BV(\mathbb{R}^n)^*$ is an open problem, but I'm not sure if this includes the case $n = 1$. If so, I'm hoping that my question might still have a known answer that doesn't depend on a characterization of the dual first.

$\DeclareMathOperator\BV{BV}$A Banach algebra $A$ is called Arens regular if the two canonical multiplications on the double dual $A^{**}$ coincide. Let $\BV(\mathbb{R})$ denote the Banach algebra of functions $\mathbb{R} \to \mathbb{C}$ which are continuous and of bounded variation.

Is $\BV(\mathbb{R})$ Arens regular?

Here's how far I've gotten until now:

• I have not seen the question mentioned in any of the literature so far. In particular, it doesn't seem to be in Palmer's book.

• Since $C^*$-algebras are Arens regular, it would be enough to realize $\BV(\mathbb{R})$ as a subalgebra of a $C^*$-algebra, or equivalently to represent it on a Hilbert space.

• Characterizing the dual space $\BV(\mathbb{R}^n)^*$ is an open problem, but I'm not sure if this includes the case $n = 1$. If so, I'm hoping that my question might still have a known answer that doesn't depend on a characterization of the dual first.